Bessel's equation arises in problems in heat conduction and mass diffusion for problems with cylindrical symmetry (and many other situations). It's solution is a linear combination of two special functions called Bessel functions of the first kind, BesselJ in Mathematica, and Bessel functions of the second kind, BesselY in Mathematica.

As an example of an application where the Bessel functions arise, consider radially-symmetric heat flow in an infinitely long cylinder of radius a that has an initial temperature profile T =T(r) and a surface temperature T(r = a) = Tr,and substitute T= &ExponentialE;-κα2&InvisibleTimes;tτ into the heat equation &PartialD;T&PartialD;t=κ&Del;2T in cylindrical coordinates. The function τ must obeyd2&InvisibleTimes;τ&PartialD;r2+ 1rdτdr+ α2τ= 0, which has the form of Bessel's equation of order ν = zero. The solution will have the form T= A J0(αr)&ExponentialE;-κα2&InvisibleTimes;t where A is a constant and J0(αr)is the Bessel function of order zero of the first kind. The parameter α must, from boundary conditions, be a solution to the equation J(αa) = 0. The initial temperature distribution, f(r), can be expanded as a series of Bessel functions of zero order, analogous to what we did with Fourier series. The general solution to the heat-flow problem will then be an infinite series of Bessel functions, each modified by an exponentially decaying amplitude, taking the form T= &Sum;n=1∞AnJ0(αnr)&ExponentialE;-καn2&InvisibleTimes;t. For a more thorough discussion, see H.S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids, Oxford University Press, Second Edition, pp. 194–196 (1959).

Legendre's equation

Legendre's equation is a second-order ODE having the following form, with the parameters m and n being integers such that n is positive and -n ≤ m ≤ n (seems to suggest applications to quantum numbers!). Legendre's equation arises in physical problems with spherical symmetry. The simplest form of the equation has only one parameter, n and takes the form:

Using Special Functions to visualize the eigensolutions for an electron bound to a fixed proton--The Hydrogen Atom

Note: There are some issues that will be updated about the following in future versions---users should not consider the following to be sufficiently checked for accuracy (yet).

Eigenfunctions for Hydrogen:

Preliminary Definitions:n will be the first quantum number n =1,2,3L will be the second quantum number L=0,1,2,... (n-1)m will be the third quantum number m = -L,-L+1,... 0,1,2, +LThese rules come from restrictions on the special functions that make up the eigenfunctions for the H-atom

In[165]:=

The eigenfunctions for the Hydrogen Atom can now be written down. The eigenfunctions will relate to the above definitions and involve the Spherical Harmonics which are related to Legendre functions and its derivatives.